A plane elastic wave `xi=a e ^(-gammax)cos( omega t - kx)`, where` a, gamma, omega, ` and `k` are constants , propagates in a homogeneous medium. Find the phase difference between the oscillations at the points where the particles, displacement amplitudes differ by `eta=1.0 %, ` if `gamma=0.42 m ^(-1)` and the wavelength is `lambda=50 cm`.

Explanation

In the given equation the particle's displacement amplitude `= a d^(-gammax)`
Let two points `x_(1)` and `x_(2)` , between which the displacement amplitude differe by `eta=1%`
So, `a e ^(- gamma x_(1))-ae^(- gammax_(2))= etaae^(-gammax_(1))`
or `e^(- gammax_(1)) (1-eta)=e^(-gammax_(2))`
or `1n(1-eta)-gamma x_(1)=-gammax_(2)`
or, `x_(2)-x_(1)=-(1n(1-eta))/( gamma) `
So path difference `=-(1n(1-n))/(gamma)`
and phase difference `=(2pi)/( lambda) xx` path difference
`=-(2pi )/( lambda)(1n( 1-eta))/(gamma)~= ( 2pieta)/( lambda. gamma)=0.3 rad`